A mathematical model for a two-pathogen, one-tick, one-host system is presented and explored. The goal of this model is to determine how long an invading pathogen persists within a tick population in which a resident pathogen is already established. The numerical simulations of the model demonstrate the parameter ranges that allow for coexistence of the two pathogens. Sensitivity analysis highlights the importance of vector-borne, tick-to-host, transmission rates on the invasion reproductive number and persistence of the pathogens over time. The model is then applied to a case study based on a reclaimed swampland field site in southeastern Virginia using field and laboratory data. The results pinpoint the thresholds required for persistence of both pathogens in the local tick population. However, the invading pathogen is not predicted to persist beyond three years. Understanding the persistence and coexistence of tick-borne pathogens will allow public health officials increased insight into tick-borne disease dynamics.

West Nile virus (WNV) is a vector-borne disease spread primarily between birds and mosquitoes. In the United States, the main methods used to control WNV are aimed at lowering the vector population through reducing births (larvicide) or increasing the mortality of adults (adulticide). We investigate the potential of a novel control, vaccination of newborn birds (nestlings). We first apply standard optimal control (OC) methodology to a nonautonomous differential equation model for WNV transmission incorporating avian stage structure and allowing larvicide, adulticide, and nestling vaccination. We convert the continuous OC results to practical advice that accounts for restrictions on the frequency of control application in the field. We then develop an alternative optimization algorithm using genetic algorithms and compare control advice for various budget constraints. We find that vaccination is recommended, provided budgets are sufficiently high, and explore how differences in control efficacy and vector biting rates on host stages affect our results. Recommendations for Resource Managers: Genetic algorithms are able to incorporate practical restrictions on control application as well as budget constraints. Advice depends on the efficacies of controls as well as model parameters. At high-efficacy estimates, adulticide is recommended for low budgets. Vaccination is recommended when budgets are sufficiently high, and sooner for scenarios where nestling hosts have increased vector exposure. If vector control has low efficacy and nestlings have increased vector exposure relative to older hosts, including vaccination may reduce levels of infectious mosquitoes below those resulting from high-efficacy vector control alone.

Researchers using ordinary differential equations to model phenomena face two main challenges among others: implementing the appropriate model and optimizing the parameters of the selected model. The latter often proves difficult or computationally expensive. Here, we implement Particle Swarm Optimization, which draws inspiration from the optimizing behavior of insect swarms in nature, as it is a simple and efficient method for fitting models to data. We demonstrate its efficacy by showing that it outstrips evolutionary computing methods previously used to analyze an epidemic model.

The use of systems of differential equations in mathematical modeling in conjunction with epidemiology continues to be an area of focused research. This paper briefly acquaints readers with epidemiology, cholera, and the need for effective control strategies; discusses cholera dynamics through a variation on the SIR epidemiological model in which two separate age classes exist in a population; finds the numeric value for R0 to be approximately 1.54 using estimated parameters for Bangladesh; and employs an optimal control resulting in a suggestion that a protection control be implemented at the end of the monsoon season.

This article provides an overview of the Akaike and Bayesian Information Criteria as applied to the setting of deterministic modelling, with the perspective that this may be a useful tool for biomathematics researchers whose primary interests lie in the analysis of compartmental models. We additionally examine a wide range mechanistic and parameter assumptions in the cholera literature through the unifying lens of model selection criteria. Five models for cholera are considered using multiple model selection formulations, and implications for cholera modelling and for model selection criteria are discussed.

We consider the problem of using time-series data to inform a corresponding deterministic model and introduce the concept of genetic algorithms (GA) as a tool for parameter estimation, providing instructions for an implementation of the method that does not require access to special toolboxes or software. We give as an example a model for cholera, a disease for which there is much mechanistic uncertainty in the literature. We use GA to find parameter sets using available time-series data from the introduction of cholera in Haiti and we discuss the value of comparing multiple parameter sets with similar performances in describing the data.

Occasional outbreaks of cholera epidemics across the world demonstrate that the disease continues to pose a public health threat. Traditional models for the spread of infectious diseases are based on systems of ordinary differential equations. Since disease dynamics such as vaccine efficacy and the risk for contracting cholera depend on the age of the humans, an age-structured model offers additional insights and the possibility of studying the effects of treatment options. The model investigated is given as a system of hyperbolic (first-order) partial differential equations in combination with ordinary differential equations. First, using a representation from the method of characteristics and a fixed point argument, we prove the existence and uniqueness of a solution to our nonlinear system. Then we present a finite difference approximation to the model and study the effect of high and low rates of shedding of cholera vibrios on the dynamics of the spread of the disease. The simulations demonstrate the explosive nature of cholera outbreaks that is observed in reality. The contrast of results for high and low rates of shedding of vibrios suggest a possible underlying cause for this effect.

Tick-borne diseases have been on the rise recently, and correspondingly, there is an increased interest in implementing control measures to decrease the risk. Optimal control provides an ideal tool to identify the best method for reducing risk while accounting for the associated costs. Using a previously published model, a variety of frameworks are assessed to identify the key factors influencing mitigation strategies. The level and duration of tick-reducing efforts are key metrics for understanding the successful reduction in tick-borne disease incidence. The results show that the punctuated nature of the tick's life history plays a critical role in reducing risk without the need for a permanent treatment programme. This work suggests that across a variety of optimal control frameworks and objective functionals within a closed environment, similar strategies are created, all suggesting that the tick-borne disease risk can be reduced to near zero without completely eliminating the tick population.

Human monocytic ehrlichiosis (Ehrlichia chaffeensis), or HME, is a tick-transmitted, ricksettisal disease with growing impact in the United States. Risk of a tick-borne disease such as HME to humans can be estimated using the prevalence of that disease in the tick population. A deterministic model for HME is explored to investigate the underlying dynamics of prevalence in tick populations, particularly when spatial considerations are allowed. The dynamics of HME in a single spatial patch are considered first to determine which model components are most important to predicting disease dynamics in a local ecology. The model is then expanded to spatially-explicit patches on which patch connectivity, the surrounding environment and boundary effects are studied. The results of this investigation show that predicting risk of this disease to humans is determined by many complicated interactions. Areas that would be endemic in isolation may or may not sustain the disease depending on the surrounding habitat. Similarly, control efforts are shown to be far more effective when applied in wooded habitats than in neighboring grassy habitats. Boundary assumptions which describe the reality of increasing habitat fragmentation additionally play a large role in predicting the endemicity of an HME outbreak. Overall, HME and all tick-borne diseases are complex, nonlinear systems that have just begun to be explored.

While cholera has been a recognized disease for two centuries, there is no strategy for its effective control. We formulate a mathematical model to include essential components such as a hyperinfectious, short-lived bacterial state, a separate class for mild human infections, and waning disease immunity. A new result quantifies contributions to the basic reproductive number from multiple infectious classes. Using optimal control theory, parameter sensitivity analysis, and numerical simulations, a cost-effective balance of multiple intervention methods is compared for two endemic populations. Results provide a framework for designing cost-effective strategies for diseases with multiple intervention methods.

Mathematical models provide a powerful tool for investigating the dynamics and control of infectious diseases, but quantifying the underlying epidemic structure can be challenging especially for new and under-studied diseases. Variations of standard SIR,SIRS,and SEIR epidemiological models are considered to determine the sensitivity of these models to various parameter values that may not be fully known when the models are used to investigate emerging diseases. Optimal control theory is applied to suggest the most effective mitigation strategy to minimize the number of individuals who become infected in the course of an infection while efficiently balancing vaccination and treatment applied to the models with various cost scenarios. The optimal control simulations suggest that regardless of the particular epidemiological structure and of the comparative cost of mitigation strategies, vaccination, if available, would be a crucial piece of any intervention plan.